My question is about $||u_k||_{L^2(\Omega)}$, where $\Omega$ is any region localized away from the closed orbit $\gamma:=\{y=0,x\in S^1\}$ i.e. $\Omega\cup\gamma=\emptyset$. How would I find the value of $\rho$ such that

$||u_k||_{L^2(\Omega)}\leq Ck^{-\rho}$?

I suspect that $\rho>0$ is non-trivial. Any ideas how I can compute the value of $\rho$?